17 |
@c $x$ in TeX, @samp{x} otherwise. |
@c $x$ in TeX, @samp{x} otherwise. |
18 |
|
|
19 |
@iftex |
@iftex |
20 |
@macro texline{stuff} |
@macro texline |
|
\stuff\ |
|
21 |
@end macro |
@end macro |
22 |
@alias infoline=comment |
@alias infoline=comment |
23 |
@tex |
@alias expr=math |
24 |
\gdef\exprsetup{\tex \let\t\ttfont \turnoffactive} |
@alias tfn=code |
|
\gdef\expr{\exprsetup$\exprfinish} |
|
|
\gdef\exprfinish#1{#1$\endgroup} |
|
|
@end tex |
|
25 |
@alias mathit=expr |
@alias mathit=expr |
26 |
@macro cpi{} |
@macro cpi{} |
27 |
@math{@pi{}} |
@math{@pi{}} |
37 |
\stuff\ |
\stuff\ |
38 |
@end macro |
@end macro |
39 |
@alias expr=samp |
@alias expr=samp |
40 |
|
@alias tfn=t |
41 |
@alias mathit=i |
@alias mathit=i |
42 |
@macro cpi{} |
@macro cpi{} |
43 |
@expr{pi} |
@expr{pi} |
660 |
``buttons'' using your left mouse button. |
``buttons'' using your left mouse button. |
661 |
|
|
662 |
@noindent |
@noindent |
663 |
Click on @key{PI}, @key{2}, and @t{y^x}. |
Click on @key{PI}, @key{2}, and @tfn{y^x}. |
664 |
|
|
665 |
@noindent |
@noindent |
666 |
Click on @key{INV}, then @key{ENTER} to swap the two results. |
Click on @key{INV}, then @key{ENTER} to swap the two results. |
9068 |
@starindex |
@starindex |
9069 |
@end ignore |
@end ignore |
9070 |
@tindex nterms |
@tindex nterms |
9071 |
If @expr{x} is the sum @expr{a + b}, then `@t{nterms(}@var{x}@t{)}' must |
If @expr{x} is the sum @expr{a + b}, then `@tfn{nterms(}@var{x}@tfn{)}' must |
9072 |
be `@t{nterms(}@var{a}@t{)}' plus `@t{nterms(}@var{b}@t{)}'. If @expr{x} |
be `@tfn{nterms(}@var{a}@tfn{)}' plus `@tfn{nterms(}@var{b}@tfn{)}'. If @expr{x} |
9073 |
is not a sum, then `@t{nterms(}@var{x}@t{)}' = 1. |
is not a sum, then `@tfn{nterms(}@var{x}@tfn{)}' = 1. |
9074 |
|
|
9075 |
@smallexample |
@smallexample |
9076 |
@group |
@group |
10869 |
notation; @pxref{Complex Formats}. |
notation; @pxref{Complex Formats}. |
10870 |
|
|
10871 |
Polar complex numbers are displayed in the form |
Polar complex numbers are displayed in the form |
10872 |
@texline `@t{(}@var{r}@t{;}@math{\theta}@t{)}' |
@texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}' |
10873 |
@infoline `@t{(}@var{r}@t{;}@var{theta}@t{)}' |
@infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}' |
10874 |
where @var{r} is the nonnegative magnitude and |
where @var{r} is the nonnegative magnitude and |
10875 |
@texline @math{\theta} |
@texline @math{\theta} |
10876 |
@infoline @var{theta} |
@infoline @var{theta} |
11287 |
A @dfn{modulo form} is a real number which is taken modulo (i.e., within |
A @dfn{modulo form} is a real number which is taken modulo (i.e., within |
11288 |
an integer multiple of) some value @var{M}. Arithmetic modulo @var{M} |
an integer multiple of) some value @var{M}. Arithmetic modulo @var{M} |
11289 |
often arises in number theory. Modulo forms are written |
often arises in number theory. Modulo forms are written |
11290 |
`@var{a} @t{mod} @var{M}', |
`@var{a} @tfn{mod} @var{M}', |
11291 |
where @var{a} and @var{M} are real numbers or HMS forms, and |
where @var{a} and @var{M} are real numbers or HMS forms, and |
11292 |
@texline @math{0 \le a < M}. |
@texline @math{0 \le a < M}. |
11293 |
@infoline @expr{0 <= a < @var{M}}. |
@infoline @expr{0 <= a < @var{M}}. |
11311 |
actually computing the power and then reducing.) |
actually computing the power and then reducing.) |
11312 |
|
|
11313 |
@cindex Modulo division |
@cindex Modulo division |
11314 |
Two modulo forms `@var{a} @t{mod} @var{M}' and `@var{b} @t{mod} @var{M}' |
Two modulo forms `@var{a} @tfn{mod} @var{M}' and `@var{b} @tfn{mod} @var{M}' |
11315 |
can be divided if @expr{a}, @expr{b}, and @expr{M} are all |
can be divided if @expr{a}, @expr{b}, and @expr{M} are all |
11316 |
integers. The result is the modulo form which, when multiplied by |
integers. The result is the modulo form which, when multiplied by |
11317 |
`@var{b} @t{mod} @var{M}', produces `@var{a} @t{mod} @var{M}'. If |
`@var{b} @tfn{mod} @var{M}', produces `@var{a} @tfn{mod} @var{M}'. If |
11318 |
there is no solution to this equation (which can happen only when |
there is no solution to this equation (which can happen only when |
11319 |
@expr{M} is non-prime), or if any of the arguments are non-integers, the |
@expr{M} is non-prime), or if any of the arguments are non-integers, the |
11320 |
division is left in symbolic form. Other operations, such as square |
division is left in symbolic form. Other operations, such as square |
11321 |
roots, are not yet supported for modulo forms. (Note that, although |
roots, are not yet supported for modulo forms. (Note that, although |
11322 |
@w{`@t{(}@var{a} @t{mod} @var{M}@t{)^.5}'} will compute a ``modulo square root'' |
@w{`@tfn{(}@var{a} @tfn{mod} @var{M}@tfn{)^.5}'} will compute a ``modulo square root'' |
11323 |
in the sense of reducing |
in the sense of reducing |
11324 |
@texline @math{\sqrt a} |
@texline @math{\sqrt a} |
11325 |
@infoline @expr{sqrt(a)} |
@infoline @expr{sqrt(a)} |
11371 |
@cindex Standard deviations |
@cindex Standard deviations |
11372 |
An @dfn{error form} is a number with an associated standard |
An @dfn{error form} is a number with an associated standard |
11373 |
deviation, as in @samp{2.3 +/- 0.12}. The notation |
deviation, as in @samp{2.3 +/- 0.12}. The notation |
11374 |
@texline `@var{x} @t{+/-} @math{\sigma}' |
@texline `@var{x} @tfn{+/-} @math{\sigma}' |
11375 |
@infoline `@var{x} @t{+/-} sigma' |
@infoline `@var{x} @tfn{+/-} sigma' |
11376 |
stands for an uncertain value which follows |
stands for an uncertain value which follows |
11377 |
a normal or Gaussian distribution of mean @expr{x} and standard |
a normal or Gaussian distribution of mean @expr{x} and standard |
11378 |
deviation or ``error'' |
deviation or ``error'' |
11418 |
of standard deviations. Actual errors often are neither Gaussian-distributed |
of standard deviations. Actual errors often are neither Gaussian-distributed |
11419 |
nor uncorrelated, and the above formulas are valid only when errors |
nor uncorrelated, and the above formulas are valid only when errors |
11420 |
are small. As an example, the error arising from |
are small. As an example, the error arising from |
11421 |
@texline `@t{sin(}@var{x} @t{+/-} @math{\sigma}@t{)}' |
@texline `@tfn{sin(}@var{x} @tfn{+/-} @math{\sigma}@tfn{)}' |
11422 |
@infoline `@t{sin(}@var{x} @t{+/-} @var{sigma}@t{)}' |
@infoline `@tfn{sin(}@var{x} @tfn{+/-} @var{sigma}@tfn{)}' |
11423 |
is |
is |
11424 |
@texline `@math{\sigma} @t{abs(cos(}@var{x}@t{))}'. |
@texline `@math{\sigma} @tfn{abs(cos(}@var{x}@tfn{))}'. |
11425 |
@infoline `@var{sigma} @t{abs(cos(}@var{x}@t{))}'. |
@infoline `@var{sigma} @tfn{abs(cos(}@var{x}@tfn{))}'. |
11426 |
When @expr{x} is close to zero, |
When @expr{x} is close to zero, |
11427 |
@texline @math{\cos x} |
@texline @math{\cos x} |
11428 |
@infoline @expr{cos(x)} |
@infoline @expr{cos(x)} |
11554 |
While it may seem that intervals and error forms are similar, they are |
While it may seem that intervals and error forms are similar, they are |
11555 |
based on entirely different concepts of inexact quantities. An error |
based on entirely different concepts of inexact quantities. An error |
11556 |
form |
form |
11557 |
@texline `@var{x} @t{+/-} @math{\sigma}' |
@texline `@var{x} @tfn{+/-} @math{\sigma}' |
11558 |
@infoline `@var{x} @t{+/-} @var{sigma}' |
@infoline `@var{x} @tfn{+/-} @var{sigma}' |
11559 |
means a variable is random, and its value could |
means a variable is random, and its value could |
11560 |
be anything but is ``probably'' within one |
be anything but is ``probably'' within one |
11561 |
@texline @math{\sigma} |
@texline @math{\sigma} |
11562 |
@infoline @var{sigma} |
@infoline @var{sigma} |
11563 |
of the mean value @expr{x}. An interval |
of the mean value @expr{x}. An interval |
11564 |
`@t{[}@var{a} @t{..@:} @var{b}@t{]}' means a |
`@tfn{[}@var{a} @tfn{..@:} @var{b}@tfn{]}' means a |
11565 |
variable's value is unknown, but guaranteed to lie in the specified |
variable's value is unknown, but guaranteed to lie in the specified |
11566 |
range. Error forms are statistical or ``average case'' approximations; |
range. Error forms are statistical or ``average case'' approximations; |
11567 |
interval arithmetic tends to produce ``worst case'' bounds on an |
interval arithmetic tends to produce ``worst case'' bounds on an |
12739 |
default simplifications for all formulas. This includes many easy and |
default simplifications for all formulas. This includes many easy and |
12740 |
fast algebraic simplifications such as @expr{a+0} to @expr{a}, and |
fast algebraic simplifications such as @expr{a+0} to @expr{a}, and |
12741 |
@expr{a + 2 a} to @expr{3 a}, as well as evaluating functions like |
@expr{a + 2 a} to @expr{3 a}, as well as evaluating functions like |
12742 |
@texline @t{deriv}@expr{(x^2,x)} |
@expr{@tfn{deriv}(x^2, x)} to @expr{2 x}. |
|
@infoline @expr{@t{deriv}(x^2, x)} |
|
|
to @expr{2 x}. |
|
12743 |
|
|
12744 |
@kindex m B |
@kindex m B |
12745 |
@pindex calc-bin-simplify-mode |
@pindex calc-bin-simplify-mode |
16507 |
|
|
16508 |
@cindex Fractional part of a number |
@cindex Fractional part of a number |
16509 |
To compute the fractional part of a number (i.e., the amount which, when |
To compute the fractional part of a number (i.e., the amount which, when |
16510 |
added to `@t{floor(}@var{n}@t{)}', will produce @var{n}) just take @var{n} |
added to `@tfn{floor(}@var{n}@tfn{)}', will produce @var{n}) just take @var{n} |
16511 |
modulo 1 using the @code{%} command. |
modulo 1 using the @code{%} command. |
16512 |
|
|
16513 |
Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm), |
Note also the @kbd{\} (integer quotient), @kbd{f I} (integer logarithm), |
16534 |
The @kbd{G} (@code{calc-argument}) [@code{arg}] command computes the |
The @kbd{G} (@code{calc-argument}) [@code{arg}] command computes the |
16535 |
``argument'' or polar angle of a complex number. For a number in polar |
``argument'' or polar angle of a complex number. For a number in polar |
16536 |
notation, this is simply the second component of the pair |
notation, this is simply the second component of the pair |
16537 |
@texline `@t{(}@var{r}@t{;}@math{\theta}@t{)}'. |
@texline `@tfn{(}@var{r}@tfn{;}@math{\theta}@tfn{)}'. |
16538 |
@infoline `@t{(}@var{r}@t{;}@var{theta}@t{)}'. |
@infoline `@tfn{(}@var{r}@tfn{;}@var{theta}@tfn{)}'. |
16539 |
The result is expressed according to the current angular mode and will |
The result is expressed according to the current angular mode and will |
16540 |
be in the range @mathit{-180} degrees (exclusive) to @mathit{+180} degrees |
be in the range @mathit{-180} degrees (exclusive) to @mathit{+180} degrees |
16541 |
(inclusive), or the equivalent range in radians. |
(inclusive), or the equivalent range in radians. |
18803 |
are both real numbers, the result uses a Gaussian distribution with mean |
are both real numbers, the result uses a Gaussian distribution with mean |
18804 |
@var{m} and standard deviation |
@var{m} and standard deviation |
18805 |
@texline @math{\sigma}. |
@texline @math{\sigma}. |
18806 |
@var{s}. |
@infoline @var{s}. |
18807 |
|
|
18808 |
If @expr{M} is an interval form, the lower and upper bounds specify the |
If @expr{M} is an interval form, the lower and upper bounds specify the |
18809 |
acceptable limits of the random numbers. If both bounds are integers, |
acceptable limits of the random numbers. If both bounds are integers, |
20483 |
weight is completely negligible.) |
weight is completely negligible.) |
20484 |
|
|
20485 |
This function also works for distributions (error forms or |
This function also works for distributions (error forms or |
20486 |
intervals). The mean of an error form `@var{a} @t{+/-} @var{b}' is simply |
intervals). The mean of an error form `@var{a} @tfn{+/-} @var{b}' is simply |
20487 |
@expr{a}. The mean of an interval is the mean of the minimum |
@expr{a}. The mean of an interval is the mean of the minimum |
20488 |
and maximum values of the interval. |
and maximum values of the interval. |
20489 |
|
|
22255 |
back on. |
back on. |
22256 |
|
|
22257 |
The most basic default simplification is the evaluation of functions. |
The most basic default simplification is the evaluation of functions. |
22258 |
For example, @expr{2 + 3} is evaluated to @expr{5}, and @expr{@t{sqrt}(9)} |
For example, @expr{2 + 3} is evaluated to @expr{5}, and @expr{@tfn{sqrt}(9)} |
22259 |
is evaluated to @expr{3}. Evaluation does not occur if the arguments |
is evaluated to @expr{3}. Evaluation does not occur if the arguments |
22260 |
to a function are somehow of the wrong type @expr{@t{tan}([2,3,4])}), |
to a function are somehow of the wrong type @expr{@tfn{tan}([2,3,4])}), |
22261 |
range (@expr{@t{tan}(90)}), or number (@expr{@t{tan}(3,5)}), |
range (@expr{@tfn{tan}(90)}), or number (@expr{@tfn{tan}(3,5)}), |
22262 |
or if the function name is not recognized (@expr{@t{f}(5)}), or if |
or if the function name is not recognized (@expr{@tfn{f}(5)}), or if |
22263 |
Symbolic mode (@pxref{Symbolic Mode}) prevents evaluation |
Symbolic mode (@pxref{Symbolic Mode}) prevents evaluation |
22264 |
(@expr{@t{sqrt}(2)}). |
(@expr{@tfn{sqrt}(2)}). |
22265 |
|
|
22266 |
Calc simplifies (evaluates) the arguments to a function before it |
Calc simplifies (evaluates) the arguments to a function before it |
22267 |
simplifies the function itself. Thus @expr{@t{sqrt}(5+4)} is |
simplifies the function itself. Thus @expr{@tfn{sqrt}(5+4)} is |
22268 |
simplified to @expr{@t{sqrt}(9)} before the @code{sqrt} function |
simplified to @expr{@tfn{sqrt}(9)} before the @code{sqrt} function |
22269 |
itself is applied. There are very few exceptions to this rule: |
itself is applied. There are very few exceptions to this rule: |
22270 |
@code{quote}, @code{lambda}, and @code{condition} (the @code{::} |
@code{quote}, @code{lambda}, and @code{condition} (the @code{::} |
22271 |
operator) do not evaluate their arguments, @code{if} (the @code{? :} |
operator) do not evaluate their arguments, @code{if} (the @code{? :} |
22390 |
@texline @math{x^{a+b}} |
@texline @math{x^{a+b}} |
22391 |
@infoline @expr{x^(a+b)} |
@infoline @expr{x^(a+b)} |
22392 |
where @expr{a} is a number, or an implicit 1 (as in @expr{x}), |
where @expr{a} is a number, or an implicit 1 (as in @expr{x}), |
22393 |
or the implicit one-half of @expr{@t{sqrt}(x)}, and similarly for |
or the implicit one-half of @expr{@tfn{sqrt}(x)}, and similarly for |
22394 |
@expr{b}. The result is written using @samp{sqrt} or @samp{1/sqrt} |
@expr{b}. The result is written using @samp{sqrt} or @samp{1/sqrt} |
22395 |
if the sum of the powers is @expr{1/2} or @expr{-1/2}, respectively. |
if the sum of the powers is @expr{1/2} or @expr{-1/2}, respectively. |
22396 |
If the sum of the powers is zero, the product is simplified to |
If the sum of the powers is zero, the product is simplified to |
22481 |
is not.) @xref{Declarations}, for ways to inform Calc that your |
is not.) @xref{Declarations}, for ways to inform Calc that your |
22482 |
variables satisfy these requirements. |
variables satisfy these requirements. |
22483 |
|
|
22484 |
As a special case of this rule, @expr{@t{sqrt}(x)^n} is simplified to |
As a special case of this rule, @expr{@tfn{sqrt}(x)^n} is simplified to |
22485 |
@texline @math{x^{n/2}} |
@texline @math{x^{n/2}} |
22486 |
@infoline @expr{x^(n/2)} |
@infoline @expr{x^(n/2)} |
22487 |
only for even integers @expr{n}. |
only for even integers @expr{n}. |
22488 |
|
|
22489 |
If @expr{a} is known to be real, @expr{b} is an even integer, and |
If @expr{a} is known to be real, @expr{b} is an even integer, and |
22490 |
@expr{c} is a half- or quarter-integer, then @expr{(a^b)^c} is |
@expr{c} is a half- or quarter-integer, then @expr{(a^b)^c} is |
22491 |
simplified to @expr{@t{abs}(a^(b c))}. |
simplified to @expr{@tfn{abs}(a^(b c))}. |
22492 |
|
|
22493 |
Also, @expr{(-a)^b} is simplified to @expr{a^b} if @expr{b} is an |
Also, @expr{(-a)^b} is simplified to @expr{a^b} if @expr{b} is an |
22494 |
even integer, or to @expr{-(a^b)} if @expr{b} is an odd integer, |
even integer, or to @expr{-(a^b)} if @expr{b} is an odd integer, |
22495 |
for any negative-looking expression @expr{-a}. |
for any negative-looking expression @expr{-a}. |
22496 |
|
|
22497 |
Square roots @expr{@t{sqrt}(x)} generally act like one-half powers |
Square roots @expr{@tfn{sqrt}(x)} generally act like one-half powers |
22498 |
@texline @math{x^{1:2}} |
@texline @math{x^{1:2}} |
22499 |
@infoline @expr{x^1:2} |
@infoline @expr{x^1:2} |
22500 |
for the purposes of the above-listed simplifications. |
for the purposes of the above-listed simplifications. |
22505 |
is changed to |
is changed to |
22506 |
@texline @math{x^{-1:2}}, |
@texline @math{x^{-1:2}}, |
22507 |
@infoline @expr{x^(-1:2)}, |
@infoline @expr{x^(-1:2)}, |
22508 |
but @expr{1 / @t{sqrt}(x)} is left alone. |
but @expr{1 / @tfn{sqrt}(x)} is left alone. |
22509 |
|
|
22510 |
@tex |
@tex |
22511 |
\bigskip |
\bigskip |
22512 |
@end tex |
@end tex |
22513 |
|
|
22514 |
Generic identity matrices (@pxref{Matrix Mode}) are simplified by the |
Generic identity matrices (@pxref{Matrix Mode}) are simplified by the |
22515 |
following rules: @expr{@t{idn}(a) + b} to @expr{a + b} if @expr{b} |
following rules: @expr{@tfn{idn}(a) + b} to @expr{a + b} if @expr{b} |
22516 |
is provably scalar, or expanded out if @expr{b} is a matrix; |
is provably scalar, or expanded out if @expr{b} is a matrix; |
22517 |
@expr{@t{idn}(a) + @t{idn}(b)} to @expr{@t{idn}(a + b)}; |
@expr{@tfn{idn}(a) + @tfn{idn}(b)} to @expr{@tfn{idn}(a + b)}; |
22518 |
@expr{-@t{idn}(a)} to @expr{@t{idn}(-a)}; @expr{a @t{idn}(b)} to |
@expr{-@tfn{idn}(a)} to @expr{@tfn{idn}(-a)}; @expr{a @tfn{idn}(b)} to |
22519 |
@expr{@t{idn}(a b)} if @expr{a} is provably scalar, or to @expr{a b} |
@expr{@tfn{idn}(a b)} if @expr{a} is provably scalar, or to @expr{a b} |
22520 |
if @expr{a} is provably non-scalar; @expr{@t{idn}(a) @t{idn}(b)} to |
if @expr{a} is provably non-scalar; @expr{@tfn{idn}(a) @tfn{idn}(b)} to |
22521 |
@expr{@t{idn}(a b)}; analogous simplifications for quotients involving |
@expr{@tfn{idn}(a b)}; analogous simplifications for quotients involving |
22522 |
@code{idn}; and @expr{@t{idn}(a)^n} to @expr{@t{idn}(a^n)} where |
@code{idn}; and @expr{@tfn{idn}(a)^n} to @expr{@tfn{idn}(a^n)} where |
22523 |
@expr{n} is an integer. |
@expr{n} is an integer. |
22524 |
|
|
22525 |
@tex |
@tex |
22528 |
|
|
22529 |
The @code{floor} function and other integer truncation functions |
The @code{floor} function and other integer truncation functions |
22530 |
vanish if the argument is provably integer-valued, so that |
vanish if the argument is provably integer-valued, so that |
22531 |
@expr{@t{floor}(@t{round}(x))} simplifies to @expr{@t{round}(x)}. |
@expr{@tfn{floor}(@tfn{round}(x))} simplifies to @expr{@tfn{round}(x)}. |
22532 |
Also, combinations of @code{float}, @code{floor} and its friends, |
Also, combinations of @code{float}, @code{floor} and its friends, |
22533 |
and @code{ffloor} and its friends, are simplified in appropriate |
and @code{ffloor} and its friends, are simplified in appropriate |
22534 |
ways. @xref{Integer Truncation}. |
ways. @xref{Integer Truncation}. |
22535 |
|
|
22536 |
The expression @expr{@t{abs}(-x)} changes to @expr{@t{abs}(x)}. |
The expression @expr{@tfn{abs}(-x)} changes to @expr{@tfn{abs}(x)}. |
22537 |
The expression @expr{@t{abs}(@t{abs}(x))} changes to |
The expression @expr{@tfn{abs}(@tfn{abs}(x))} changes to |
22538 |
@expr{@t{abs}(x)}; in fact, @expr{@t{abs}(x)} changes to @expr{x} or |
@expr{@tfn{abs}(x)}; in fact, @expr{@tfn{abs}(x)} changes to @expr{x} or |
22539 |
@expr{-x} if @expr{x} is provably nonnegative or nonpositive |
@expr{-x} if @expr{x} is provably nonnegative or nonpositive |
22540 |
(@pxref{Declarations}). |
(@pxref{Declarations}). |
22541 |
|
|
22542 |
While most functions do not recognize the variable @code{i} as an |
While most functions do not recognize the variable @code{i} as an |
22543 |
imaginary number, the @code{arg} function does handle the two cases |
imaginary number, the @code{arg} function does handle the two cases |
22544 |
@expr{@t{arg}(@t{i})} and @expr{@t{arg}(-@t{i})} just for convenience. |
@expr{@tfn{arg}(@tfn{i})} and @expr{@tfn{arg}(-@tfn{i})} just for convenience. |
22545 |
|
|
22546 |
The expression @expr{@t{conj}(@t{conj}(x))} simplifies to @expr{x}. |
The expression @expr{@tfn{conj}(@tfn{conj}(x))} simplifies to @expr{x}. |
22547 |
Various other expressions involving @code{conj}, @code{re}, and |
Various other expressions involving @code{conj}, @code{re}, and |
22548 |
@code{im} are simplified, especially if some of the arguments are |
@code{im} are simplified, especially if some of the arguments are |
22549 |
provably real or involve the constant @code{i}. For example, |
provably real or involve the constant @code{i}. For example, |
22550 |
@expr{@t{conj}(a + b i)} is changed to |
@expr{@tfn{conj}(a + b i)} is changed to |
22551 |
@expr{@t{conj}(a) - @t{conj}(b) i}, or to @expr{a - b i} if @expr{a} |
@expr{@tfn{conj}(a) - @tfn{conj}(b) i}, or to @expr{a - b i} if @expr{a} |
22552 |
and @expr{b} are known to be real. |
and @expr{b} are known to be real. |
22553 |
|
|
22554 |
Functions like @code{sin} and @code{arctan} generally don't have |
Functions like @code{sin} and @code{arctan} generally don't have |
22558 |
these functions, though. |
these functions, though. |
22559 |
|
|
22560 |
One important simplification that does occur is that |
One important simplification that does occur is that |
22561 |
@expr{@t{ln}(@t{e})} is simplified to 1, and @expr{@t{ln}(@t{e}^x)} is |
@expr{@tfn{ln}(@tfn{e})} is simplified to 1, and @expr{@tfn{ln}(@tfn{e}^x)} is |
22562 |
simplified to @expr{x} for any @expr{x}. This occurs even if you have |
simplified to @expr{x} for any @expr{x}. This occurs even if you have |
22563 |
stored a different value in the Calc variable @samp{e}; but this would |
stored a different value in the Calc variable @samp{e}; but this would |
22564 |
be a bad idea in any case if you were also using natural logarithms! |
be a bad idea in any case if you were also using natural logarithms! |
22565 |
|
|
22566 |
Among the logical functions, @t{(@var{a} <= @var{b})} changes to |
Among the logical functions, @tfn{(@var{a} <= @var{b})} changes to |
22567 |
@t{@var{a} > @var{b}} and so on. Equations and inequalities where both sides |
@tfn{@var{a} > @var{b}} and so on. Equations and inequalities where both sides |
22568 |
are either negative-looking or zero are simplified by negating both sides |
are either negative-looking or zero are simplified by negating both sides |
22569 |
and reversing the inequality. While it might seem reasonable to simplify |
and reversing the inequality. While it might seem reasonable to simplify |
22570 |
@expr{!!x} to @expr{x}, this would not be valid in general because |
@expr{!!x} to @expr{x}, this would not be valid in general because |
22688 |
Square roots of integer or rational arguments are simplified in |
Square roots of integer or rational arguments are simplified in |
22689 |
several ways. (Note that these will be left unevaluated only in |
several ways. (Note that these will be left unevaluated only in |
22690 |
Symbolic mode.) First, square integer or rational factors are |
Symbolic mode.) First, square integer or rational factors are |
22691 |
pulled out so that @expr{@t{sqrt}(8)} is rewritten as |
pulled out so that @expr{@tfn{sqrt}(8)} is rewritten as |
22692 |
@texline @math{2\,\t{sqrt}(2)}. |
@texline @math{2\,@tfn{sqrt}(2)}. |
22693 |
@infoline @expr{2 sqrt(2)}. |
@infoline @expr{2 sqrt(2)}. |
22694 |
Conceptually speaking this implies factoring the argument into primes |
Conceptually speaking this implies factoring the argument into primes |
22695 |
and moving pairs of primes out of the square root, but for reasons of |
and moving pairs of primes out of the square root, but for reasons of |
22696 |
efficiency Calc only looks for primes up to 29. |
efficiency Calc only looks for primes up to 29. |
22697 |
|
|
22698 |
Square roots in the denominator of a quotient are moved to the |
Square roots in the denominator of a quotient are moved to the |
22699 |
numerator: @expr{1 / @t{sqrt}(3)} changes to @expr{@t{sqrt}(3) / 3}. |
numerator: @expr{1 / @tfn{sqrt}(3)} changes to @expr{@tfn{sqrt}(3) / 3}. |
22700 |
The same effect occurs for the square root of a fraction: |
The same effect occurs for the square root of a fraction: |
22701 |
@expr{@t{sqrt}(2:3)} changes to @expr{@t{sqrt}(6) / 3}. |
@expr{@tfn{sqrt}(2:3)} changes to @expr{@tfn{sqrt}(6) / 3}. |
22702 |
|
|
22703 |
@tex |
@tex |
22704 |
\bigskip |
\bigskip |
22731 |
@end tex |
@end tex |
22732 |
|
|
22733 |
Trigonometric functions are simplified in several ways. First, |
Trigonometric functions are simplified in several ways. First, |
22734 |
@expr{@t{sin}(@t{arcsin}(x))} is simplified to @expr{x}, and |
@expr{@tfn{sin}(@tfn{arcsin}(x))} is simplified to @expr{x}, and |
22735 |
similarly for @code{cos} and @code{tan}. If the argument to |
similarly for @code{cos} and @code{tan}. If the argument to |
22736 |
@code{sin} is negative-looking, it is simplified to |
@code{sin} is negative-looking, it is simplified to |
22737 |
@expr{-@t{sin}(x),}, and similarly for @code{cos} and @code{tan}. |
@expr{-@tfn{sin}(x),}, and similarly for @code{cos} and @code{tan}. |
22738 |
Finally, certain special values of the argument are recognized; |
Finally, certain special values of the argument are recognized; |
22739 |
@pxref{Trigonometric and Hyperbolic Functions}. |
@pxref{Trigonometric and Hyperbolic Functions}. |
22740 |
|
|
22741 |
Trigonometric functions of inverses of different trigonometric |
Trigonometric functions of inverses of different trigonometric |
22742 |
functions can also be simplified, as in @expr{@t{sin}(@t{arccos}(x))} |
functions can also be simplified, as in @expr{@tfn{sin}(@tfn{arccos}(x))} |
22743 |
to @expr{@t{sqrt}(1 - x^2)}. |
to @expr{@tfn{sqrt}(1 - x^2)}. |
22744 |
|
|
22745 |
Hyperbolic functions of their inverses and of negative-looking |
Hyperbolic functions of their inverses and of negative-looking |
22746 |
arguments are also handled, as are exponentials of inverse |
arguments are also handled, as are exponentials of inverse |
22749 |
No simplifications for inverse trigonometric and hyperbolic |
No simplifications for inverse trigonometric and hyperbolic |
22750 |
functions are known, except for negative arguments of @code{arcsin}, |
functions are known, except for negative arguments of @code{arcsin}, |
22751 |
@code{arctan}, @code{arcsinh}, and @code{arctanh}. Note that |
@code{arctan}, @code{arcsinh}, and @code{arctanh}. Note that |
22752 |
@expr{@t{arcsin}(@t{sin}(x))} can @emph{not} safely change to |
@expr{@tfn{arcsin}(@tfn{sin}(x))} can @emph{not} safely change to |
22753 |
@expr{x}, since this only correct within an integer multiple of |
@expr{x}, since this only correct within an integer multiple of |
22754 |
@texline @math{2 \pi} |
@texline @math{2 \pi} |
22755 |
@infoline @expr{2 pi} |
@infoline @expr{2 pi} |
22756 |
radians or 360 degrees. However, @expr{@t{arcsinh}(@t{sinh}(x))} is |
radians or 360 degrees. However, @expr{@tfn{arcsinh}(@tfn{sinh}(x))} is |
22757 |
simplified to @expr{x} if @expr{x} is known to be real. |
simplified to @expr{x} if @expr{x} is known to be real. |
22758 |
|
|
22759 |
Several simplifications that apply to logarithms and exponentials |
Several simplifications that apply to logarithms and exponentials |
22760 |
are that @expr{@t{exp}(@t{ln}(x))}, |
are that @expr{@tfn{exp}(@tfn{ln}(x))}, |
22761 |
@texline @t{e}@math{^{\ln(x)}}, |
@texline @tfn{e}@math{^{\ln(x)}}, |
22762 |
@infoline @expr{e^@t{ln}(x)}, |
@infoline @expr{e^@tfn{ln}(x)}, |
22763 |
and |
and |
22764 |
@texline @math{10^{{\rm log10}(x)}} |
@texline @math{10^{{\rm log10}(x)}} |
22765 |
@infoline @expr{10^@t{log10}(x)} |
@infoline @expr{10^@tfn{log10}(x)} |
22766 |
all reduce to @expr{x}. Also, @expr{@t{ln}(@t{exp}(x))}, etc., can |
all reduce to @expr{x}. Also, @expr{@tfn{ln}(@tfn{exp}(x))}, etc., can |
22767 |
reduce to @expr{x} if @expr{x} is provably real. The form |
reduce to @expr{x} if @expr{x} is provably real. The form |
22768 |
@expr{@t{exp}(x)^y} is simplified to @expr{@t{exp}(x y)}. If @expr{x} |
@expr{@tfn{exp}(x)^y} is simplified to @expr{@tfn{exp}(x y)}. If @expr{x} |
22769 |
is a suitable multiple of |
is a suitable multiple of |
22770 |
@texline @math{\pi i} |
@texline @math{\pi i} |
22771 |
@infoline @expr{pi i} |
@infoline @expr{pi i} |
22772 |
(as described above for the trigonometric functions), then |
(as described above for the trigonometric functions), then |
22773 |
@expr{@t{exp}(x)} or @expr{e^x} will be expanded. Finally, |
@expr{@tfn{exp}(x)} or @expr{e^x} will be expanded. Finally, |
22774 |
@expr{@t{ln}(x)} is simplified to a form involving @code{pi} and |
@expr{@tfn{ln}(x)} is simplified to a form involving @code{pi} and |
22775 |
@code{i} where @expr{x} is provably negative, positive imaginary, or |
@code{i} where @expr{x} is provably negative, positive imaginary, or |
22776 |
negative imaginary. |
negative imaginary. |
22777 |
|
|
22843 |
|
|
22844 |
Inverse trigonometric or hyperbolic functions, called with their |
Inverse trigonometric or hyperbolic functions, called with their |
22845 |
corresponding non-inverse functions as arguments, are simplified |
corresponding non-inverse functions as arguments, are simplified |
22846 |
by @kbd{a e}. For example, @expr{@t{arcsin}(@t{sin}(x))} changes |
by @kbd{a e}. For example, @expr{@tfn{arcsin}(@tfn{sin}(x))} changes |
22847 |
to @expr{x}. Also, @expr{@t{arcsin}(@t{cos}(x))} and |
to @expr{x}. Also, @expr{@tfn{arcsin}(@tfn{cos}(x))} and |
22848 |
@expr{@t{arccos}(@t{sin}(x))} both change to @expr{@t{pi}/2 - x}. |
@expr{@tfn{arccos}(@tfn{sin}(x))} both change to @expr{@tfn{pi}/2 - x}. |
22849 |
These simplifications are unsafe because they are valid only for |
These simplifications are unsafe because they are valid only for |
22850 |
values of @expr{x} in a certain range; outside that range, values |
values of @expr{x} in a certain range; outside that range, values |
22851 |
are folded down to the 360-degree range that the inverse trigonometric |
are folded down to the 360-degree range that the inverse trigonometric |
22861 |
the powers cancel to get @expr{x}, which is valid for positive values |
the powers cancel to get @expr{x}, which is valid for positive values |
22862 |
of @expr{x} but not for negative or complex values. |
of @expr{x} but not for negative or complex values. |
22863 |
|
|
22864 |
Similarly, @expr{@t{sqrt}(x^a)} and @expr{@t{sqrt}(x)^a} are both |
Similarly, @expr{@tfn{sqrt}(x^a)} and @expr{@tfn{sqrt}(x)^a} are both |
22865 |
simplified (possibly unsafely) to |
simplified (possibly unsafely) to |
22866 |
@texline @math{x^{a/2}}. |
@texline @math{x^{a/2}}. |
22867 |
@infoline @expr{x^(a/2)}. |
@infoline @expr{x^(a/2)}. |
22868 |
|
|
22869 |
Forms like @expr{@t{sqrt}(1 - sin(x)^2)} are simplified to, e.g., |
Forms like @expr{@tfn{sqrt}(1 - sin(x)^2)} are simplified to, e.g., |
22870 |
@expr{@t{cos}(x)}. Calc has identities of this sort for @code{sin}, |
@expr{@tfn{cos}(x)}. Calc has identities of this sort for @code{sin}, |
22871 |
@code{cos}, @code{tan}, @code{sinh}, and @code{cosh}. |
@code{cos}, @code{tan}, @code{sinh}, and @code{cosh}. |
22872 |
|
|
22873 |
Arguments of square roots are partially factored to look for |
Arguments of square roots are partially factored to look for |
22874 |
squared terms that can be extracted. For example, |
squared terms that can be extracted. For example, |
22875 |
@expr{@t{sqrt}(a^2 b^3 + a^3 b^2)} simplifies to |
@expr{@tfn{sqrt}(a^2 b^3 + a^3 b^2)} simplifies to |
22876 |
@expr{a b @t{sqrt}(a+b)}. |
@expr{a b @tfn{sqrt}(a+b)}. |
22877 |
|
|
22878 |
The simplifications of @expr{@t{ln}(@t{exp}(x))}, |
The simplifications of @expr{@tfn{ln}(@tfn{exp}(x))}, |
22879 |
@expr{@t{ln}(@t{e}^x)}, and @expr{@t{log10}(10^x)} to @expr{x} are also |
@expr{@tfn{ln}(@tfn{e}^x)}, and @expr{@tfn{log10}(10^x)} to @expr{x} are also |
22880 |
unsafe because of problems with principal values (although these |
unsafe because of problems with principal values (although these |
22881 |
simplifications are safe if @expr{x} is known to be real). |
simplifications are safe if @expr{x} is known to be real). |
22882 |
|
|
24428 |
or all be plain numbers. Error forms can go anywhere but generally |
or all be plain numbers. Error forms can go anywhere but generally |
24429 |
go on the numbers in the last row of the data matrix. If the last |
go on the numbers in the last row of the data matrix. If the last |
24430 |
row contains error forms |
row contains error forms |
24431 |
@texline `@var{y_i}@w{ @t{+/-} }@math{\sigma_i}', |
@texline `@var{y_i}@w{ @tfn{+/-} }@math{\sigma_i}', |
24432 |
@infoline `@var{y_i}@w{ @t{+/-} }@var{sigma_i}', |
@infoline `@var{y_i}@w{ @tfn{+/-} }@var{sigma_i}', |
24433 |
then the |
then the |
24434 |
@texline @math{\chi^2} |
@texline @math{\chi^2} |
24435 |
@infoline @expr{chi^2} |
@infoline @expr{chi^2} |
24581 |
@item 2-9 |
@item 2-9 |
24582 |
Polynomials. @mathit{a + b x + c x^2 + d x^3}. |
Polynomials. @mathit{a + b x + c x^2 + d x^3}. |
24583 |
@item e |
@item e |
24584 |
Exponential. @mathit{a} @t{exp}@mathit{(b x)} @t{exp}@mathit{(c y)}. |
Exponential. @mathit{a} @tfn{exp}@mathit{(b x)} @tfn{exp}@mathit{(c y)}. |
24585 |
@item E |
@item E |
24586 |
Base-10 exponential. @mathit{a} @t{10^}@mathit{(b x)} @t{10^}@mathit{(c y)}. |
Base-10 exponential. @mathit{a} @tfn{10^}@mathit{(b x)} @tfn{10^}@mathit{(c y)}. |
24587 |
@item x |
@item x |
24588 |
Exponential (alternate notation). @t{exp}@mathit{(a + b x + c y)}. |
Exponential (alternate notation). @tfn{exp}@mathit{(a + b x + c y)}. |
24589 |
@item X |
@item X |
24590 |
Base-10 exponential (alternate). @t{10^}@mathit{(a + b x + c y)}. |
Base-10 exponential (alternate). @tfn{10^}@mathit{(a + b x + c y)}. |
24591 |
@item l |
@item l |
24592 |
Logarithmic. @mathit{a + b} @t{ln}@mathit{(x) + c} @t{ln}@mathit{(y)}. |
Logarithmic. @mathit{a + b} @tfn{ln}@mathit{(x) + c} @tfn{ln}@mathit{(y)}. |
24593 |
@item L |
@item L |
24594 |
Base-10 logarithmic. @mathit{a + b} @t{log10}@mathit{(x) + c} @t{log10}@mathit{(y)}. |
Base-10 logarithmic. @mathit{a + b} @tfn{log10}@mathit{(x) + c} @tfn{log10}@mathit{(y)}. |
24595 |
@item ^ |
@item ^ |
24596 |
General exponential. @mathit{a b^x c^y}. |
General exponential. @mathit{a b^x c^y}. |
24597 |
@item p |
@item p |
34778 |
@r{ @: . @:number @: @:@:0.number} |
@r{ @: . @:number @: @:@:0.number} |
34779 |
@r{ @: _ @:number @: @:-@:number} |
@r{ @: _ @:number @: @:-@:number} |
34780 |
@r{ @: e @:number @: @:@:1e number} |
@r{ @: e @:number @: @:@:1e number} |
34781 |
@r{ @: # @:number @: @:@:current-radix@t{#}number} |
@r{ @: # @:number @: @:@:current-radix@tfn{#}number} |
34782 |
@r{ @: P @:(in number) @: @:+/-@:} |
@r{ @: P @:(in number) @: @:+/-@:} |
34783 |
@r{ @: M @:(in number) @: @:mod@:} |
@r{ @: M @:(in number) @: @:mod@:} |
34784 |
@r{ @: @@ ' " @: (in number)@: @:@:HMS form} |
@r{ @: @@ ' " @: (in number)@: @:@:HMS form} |
35338 |
@r{ @: s & @:var @: 29,47 @:calc-store-inv@: (v^-1)} |
@r{ @: s & @:var @: 29,47 @:calc-store-inv@: (v^-1)} |
35339 |
@r{ @: s [ @:var @: 29,47 @:calc-store-decr@: (v-1)} |
@r{ @: s [ @:var @: 29,47 @:calc-store-decr@: (v-1)} |
35340 |
@r{ @: s ] @:var @: 29,47 @:calc-store-incr@: (v-(-1))} |
@r{ @: s ] @:var @: 29,47 @:calc-store-incr@: (v-(-1))} |
35341 |
@r{ a b@: s : @: @: 2 @:assign@:(a,b) a @t{:=} b} |
@r{ a b@: s : @: @: 2 @:assign@:(a,b) a @tfn{:=} b} |
35342 |
@r{ a@: s = @: @: 1 @:evalto@:(a,b) a @t{=>}} |
@r{ a@: s = @: @: 1 @:evalto@:(a,b) a @tfn{=>}} |
35343 |
|
|
35344 |
@c |
@c |
35345 |
@r{ @: t [ @: @: 4 @:calc-trail-first@:} |
@r{ @: t [ @: @: 4 @:calc-trail-first@:} |